I am trying to show that the square of any Fibonacci Number differs by one from the product of its neighbors:
My best attempt is to substitute Binet's Formula:
and simplify, but keep getting buggered near the end. I get down to:
and can go no further due to the unequal bases. I know (because even values of "n" yield -1) that:
which would make the final formula:
but I seem to be missing something. I could swear I've seen this done, but can't find it anywhere. Any pointers as to what I'm missing?